Determination of Upperbound Failure Rate by Graphic Confidence Interval Estimate

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Determination ofUpperbound Failure
RatebyGraphic Confidence Interval Estimate
LAUR-01-1671

• K. S. Kim (Kyo)
• Los Alamos National Laboratory
• Los Alamos, NM 87545
• E-mail kyokim_at_lanl.gov

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If you believe that selecting Power Ball
numbersis a random process, that is, a Poisson
process,then your chance of winning is 1 in
1000000. But considering your horoscope today
and invokingthe Bayesian theorem, your chance
can be 1 in 5.Of course, there are sampling
errors of plus-minus.
Gee, I wonder whatis the odd of gettingmy money
back
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DOE Hazard Analysis Requirement
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• DOE Order 5480.23 requires Hazard Analysis for
  all Nuclear Facilities
• Hazard Analysis entails estimation of Consequence
  and Likelihood (or Frequency) of potential
  accidents
• Potential Accidents are Binned according to
  Consequence Frequency for determination of
  further analysis and necessary Controls
• DOE-STD-3009 provides Example for Binning
• LANL Binning Matrix (risk matrix)

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LANL Binning Example
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Method for Frequency Determination
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• Historical Record of Event Occurrence (number of
  events per component-time or N/CT)
• A simple division of N/CT ignores uncertainty (1
  event in 10 component-yrs and 100 events per
  1,000 component-yrs would be represented by the
  same frequency value of 0.1/yr)
• Not useful for a type of accident that has not
  occurred yet (Zero-occurrence events)
• Fault Tree/Event Tree Method (for PRA) can be
  used for Overall Accident Likelihood Historical
  record is used for estimation of initiating event
  frequency or component failure rate/frequency

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Statistical Inference Primer
LAUR-01-1671

• Typical occurrences of failure (spill, leaks,
  fire, etc.) are considered as random discrete
  events in space and time (Poisson process), thus
  Poisson distribution can be assumed for the
  Failure Rate (or Frequency)
• Classical Confidence Intervals have the property
  that Probability of parameters of interest being
  contained within the Confidence Interval is at
  least at the specified confidence level in
  repeated samplings
• Upperbound Confidence Interval for Poisson
  process can be approximated by Chi-square
  distribution function
• ?U (1-P) is upper 100(1-P) confidence limit
  (or interval) of ?,
• P is exceedance probability,
• ?2(2N2 1-P) is chi-square distribution with
  2N2 degrees of freedom

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Chi-square Distribution
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Graphic Method
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• Zero-occurrence Events
• Nonzero-occurrence Events

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Zero-occurrence Events
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Nonzero-occurrence Events
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Examples
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• Upperbound frequency estimate of a liquid
  radwaste spill of more than 5 gallons for a
  Preliminary Hazard Analysis (desired confidence
  level is set as 80 or exceedance probability of
  0.2). No such spill has been recorded for 3
  similar facilities in 10 years.
• Upperbound frequency estimate of a fire lasting
  longer than 2 hours for Design Basis Accident
  Analysis (desired confidence level is set as 95
  or exceedance probability of 0.05). Four (4)
  such fires have been recorded in 5 similar
  facilities during a sampling period 12 years.

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Zero-occurrence Events
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(No occurrence for 3 components in 10 years, 80
Confidence Interval)
C3, T10 yr ?U (80) Z/CT 1.6/30 0.053
/yr
Spill frequency is less than 0.053/yr with 80
confidence
Z1.6
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Nonzero-occurrence Events(4 occurrences for 5
components in 12 years, 95 Confidence interval)
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N4, C5, T12 yrs ?U(95) R(N/CT)
2.30.067 0.15/yr Fire frequency is less than
0.15/yr with 95 confidence
R2.3
N4
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Concluding Remarks
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• Setting Confidence Level depends on analysts
• Higher Level for events with sparse historical
  data (infrequent or rare events)
• Higher Level for Conservative Design Analysis
  (95 for DBA)
• Lower Level for expected or best estimate
  analysis (50)

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